Pat the chicken thighs dry with paper towels. Season generously on both sides with salt and pepper. Heat 1 tbsp olive oil in your Dutch oven over medium-high heat until it shimmers. Add the chicken in a single layer and sear 4 to 5 minutes per side until deeply golden brown. Transfer to a plate (they will not be cooked through yet).
Reduce the heat to medium-low. Add a splash more oil if the pan looks dry. Add the diced onion and a pinch of salt; cook, stirring occasionally, for 7 to 8 minutes until completely soft and translucent. Add the garlic, oregano, and red pepper flakes and cook for 1 minute until fragrant.
Stir in the tomato paste and cook for 2 minutes, stirring constantly, until it darkens from bright red to a rusty brick color. Pour in the crushed tomatoes and chicken broth, scraping the bottom of the pot to lift the browned bits. Bring to a gentle simmer.
Nestle the seared chicken thighs back into the sauce, including any accumulated juices. Cover the pot, reduce the heat to low, and simmer for 15 minutes until the chicken is cooked through and tender enough to shred easily with a fork.
Remove the chicken to a cutting board. Use two forks to shred it into bite-sized pieces. Return the shredded chicken to the sauce. Stir in 1/2 cup of water or broth — the sauce needs to be slightly loose at this stage because the pasta will absorb liquid.
Meanwhile, bring a separate pot of salted water to a boil. Add the pasta and cook for 2 minutes less than the package directions indicate. Drain the pasta (reserve 1 cup of the starchy pasta water) and add it directly to the sauce. Stir well and let it simmer together for 2 to 3 minutes, stirring occasionally. If the sauce gets too thick, splash in reserved pasta water until silky.
Taste the sauce and adjust salt and pepper. Remove from heat. Stir in a generous handful of fresh basil leaves until they wilt slightly. Serve hot with a shower of grated Parmesan.